0
votes
0answers
31 views

Know a formula for $x \pmod{y + z}$?

It seems natural to want to add the $x \pmod y$ operator to the integers (or other generalization of the integers?), but perhaps it has no algebraic properties. Can you prove that there is no ...
1
vote
3answers
41 views

prove that if $a$ is not a unit in $\mathbb Z$/m$\mathbb Z$ then $a$ is a zero divisor in $\mathbb Z$/m$\mathbb Z$?

Ho can I prove that if $a$ is not a unit in $\mathbb Z$/m$\mathbb Z$ then $a$ is a zero divisor in $\mathbb Z$/m$\mathbb Z$ ? I am stuck on this problem I would appreciate a lot your help thanks!!
3
votes
1answer
50 views

Show: $\varphi\colon\mathbb{Z}_{mn}\to\mathbb{Z}_m\times\mathbb{Z}_n, k\mapsto (k\% m,k\% n)$ is a ring isomorphism for $m$ and $n$ relatively prim

Let $m\in\mathbb{Z}, n\in\mathbb{N}$. Then there exist unique elements $q\in\mathbb{Z}, r\in\mathbb{N}$ with $0\leq r<n$ and $m=qn+r$. We write $r:=m\% n$. Let $m,n\in\mathbb{N}$ be relatively ...
1
vote
1answer
30 views

Show that the solutions lifted by Hensel's lemma, are the only solutions in $\mathbb{Z} / p^k \mathbb{Z}$

Suppose that $f(x) \in \mathbb{Z}[X]$ is a polynomial to which we can apply Hensel's Lemma. Suppose the set $A$ is the set of solutions of $f(x)$ in $\mathbb{Z} / p \mathbb{Z}$. I'd like to show that ...
1
vote
0answers
69 views

Proposition 17, p. 68 of Lang's Algebraic Number Theory

I'm stuck on a detail of the following propositon: Let $K, E$ be linearly disjoint number fields with degrees $n$ and $m$ over $\mathbb{Q}$ whose discriminants (over $\mathbb{Q}$) are relatively ...
6
votes
3answers
76 views

Let $R = \mathbb Z[i]$. Show $I \cap \mathbb Z$ is an ideal in $\mathbb Z$, for all $a \in I \cap \mathbb Z$, $10 \mid a^2 = N(a)$.

Let $R = \mathbb Z[i]$, $z = 3+i$ and $I = \langle z \rangle$. I need to show $I \cap \mathbb Z$ is an ideal in $\mathbb Z$, for all $a \in I \cap \mathbb Z$, $10 \mid a^2 = N(a)$ and $10 \mid a$, ...
3
votes
1answer
23 views

Finding an $i \in \mathbb{N}$ s.t. $im + k = p$

Let $m,k \in \mathbb{N}$ s.t. $\gcd(m,k) = 1$. Let $\pi$ be a prime positive integer. Question: Does there always exist an $i \in \mathbb{N}$ s.t. $im + k = p$ s.t. $p$ is a prime positive integer ...
7
votes
1answer
60 views

Halving One in Odd Size Rings

Consider the rings $\mathbb{Z} /n \mathbb{Z}$ where $n$ is odd. Every number is even in such rings. Assume we start with $1$ and keep "halving" until we get back to $1$. What can be said about the ...
5
votes
2answers
56 views

Greatest common divisor of $2 + 3i$ and $1-i$ in $\mathbb{Z}[i]$

Greatest common divisor of $2 + 3i$ and $1-i$ in $\mathbb{Z}[i]$ Here is my attempt at solving this using a generalized Euclid's Algorithm. Does it look alright? Step 1 $2 + 3i = M(1-i) + N$ ...
1
vote
2answers
57 views

Divisors of $x^3 + x + 1$ in $Z_3[x]$

Divisors of $f(x) = x^3 + x + 1$ in $Z_3[x]$ Do I have to manually check whether each polynomial in $Z_3[x]$ with degree less than $3$ divides $f$ or is there a better way? That's $3^3 = 27$ ...
1
vote
1answer
64 views

Number of ideals in $\Bbb Z[x]/(x^3+1, 7)$

I am trying to find the number of ideals in $R:=\Bbb Z[x]/(x^3+1, 7)$ and $S:=\Bbb Z[x]/(x^3+1, 3)$. I started with $R$ and tried to write it in terms of familiar rings, by using fundamental ...
2
votes
2answers
101 views

$(\Bbb Z[x]/(x^{n+1}))^\times\cong\Bbb Z/2\Bbb Z \times \prod _{i=1}^n \Bbb Z$ [duplicate]

I'm trying to prove the group isomorphism $(\Bbb Z[x]/(x^{n+1}))^\times\cong\Bbb Z/2\Bbb Z \times \prod _{i=1}^n \Bbb Z$. Obviously I tried to establish a ring isomorphism from $\Bbb Z[x]/(x^{n+1})$ ...
-1
votes
1answer
76 views

Idempotent elements in $(\mathbb{Z}_n,+,\cdot)$ [duplicate]

Can we find the idempotents in $(\mathbb{Z}_n,+,\cdot)$ for any $n$? Is there a general rule? Note: Trying to consider the prime factors!
2
votes
0answers
93 views

In $ℤ/Nℤ$, which units are successors to zero divisors?

What are the units $x$ in $ℤ/Nℤ$ of the form $x = 1 + \overline{kd}$ for a divisor $d$ of $N$ and $k ∈ ℤ$, i.e. $$U_N[d] := \{x ∈ (ℤ/Nℤ)^×;\; ∃ k ∈ ℤ : x = 1 + \overline{kd}\} = \ker \big((ℤ/Nℤ)^× → ...
3
votes
1answer
83 views

$\mathbb{Z}/m\mathbb{Z}$: A Complete Set of Representatives

So, I'm letting ${\scr{A}}=\{a_1,\dots,a_n\}$ be a complete set of representatives (C.S.R.) for $\mathbb{Z}/m\mathbb{Z}$. I'm considering all $b\in \mathbb{Z}$ and seeing if ${\scr{B}}=\{a_1+b, a_2+b, ...
1
vote
0answers
13 views

Are there ways to get modulated values similar to cyclic and negacyclic convolutions?

The Wikipedia article on the Schönhage–Strassen algorithm states that there are methods that can get values modulo $a^n+1$ or $a^n-1$ for some value $a$. More specifically, it shows that the cyclic ...
1
vote
6answers
615 views

$ p^{\frac1n} $ is irrational if $p $ is prime and $n>1$ [closed]

How can we prove that $ p^{\frac1n} $ is irrational if $p $ is prime and $n>1$?
5
votes
2answers
163 views

Primes of the form $x^2 + y^2$ can only be written as $a^2 + b^2$ in one way?

Let $p$ be a prime number. Let $x,y,a,b$ be distinct positive integers. If $p=x^2 + y^2$ then $p\ne a^2+b^2$. Is this true ? If so why ? What are the proofs for this ? I know the Fibonacci identitity ...
3
votes
2answers
115 views

Lost on algebra notation

I'm in a basic number theory course and, never having taken college algebra, I'm lost on some of the notation. I'm wondering what some of these notations mean: 1: $\mathbb{Q}(i)$ ... I know that what ...
4
votes
5answers
215 views

Primes of the form $a^2+b^2$ : a technical point.

One can classify the prime integers $p$ which can be written as $p=a^2+b^2$ for some integers $a,b\in\mathbb{Z}$ by studying how $p$ decomposes in the ring of Gauss integers $\mathbb{Z}[i]$. Most ...
1
vote
1answer
112 views

What about the Cauchy-Frobenius-orbit-counting formula

I know the proposition that says: Let $\lambda$ be a homomorphism from a finite group $G$ into $\mathbb{C}^{\times}$. Suppose that $G$ acts on some finite set $\Omega$ and let $M$ be the number of ...
3
votes
3answers
323 views

Show $m^p+n^p\equiv 0 \mod p$ implies $m^p+n^p\equiv 0 \mod p^2$

Let $p$ an odd prime. Show that $m^p+n^p\equiv 0 \pmod p$ implies $m^p+n^p\equiv 0 \pmod{p^2}$.
2
votes
1answer
116 views

Equivalence classes on Z

So I'm given that R is an equivalence relation on Z, and that adding classes in the natural way is well defined. That is, class(a)+class(b) = class(a + b). I want to show that R can only be equality ...
2
votes
0answers
71 views

Solving linear inequalities over rings

The concrete problem: for any given $N\ge 1$ I have a system of $2^N-1$ linear inequalities over $\mathbb{Z}_6^N$ which looks like this: for every nonempty $S\subseteq[N]$ there is some ...
0
votes
3answers
396 views

zero divisors and units for the group $\mathbb{Z}/n\mathbb{Z}$ with integer $n$

given the ring $ \mathbb{Z}/n\mathbb{Z} $ is always true that $ \mathbb{Z}/n\mathbb{Z}=[\text{zero divisor}]+[\text{units}] $ how can evaluate the zero divisor and units ?? I believe that $ a x=0 ...
2
votes
2answers
151 views

an invertible element $i$ in $\mathbb Z_n$ must be coprime to $n$

Let $n$ be an integer and $i\in \{1,\cdots,n-1\}$. I want to show that $i$ is invertible in $\mathbb Z_n$ if and only if $i$ is coprime to $n$. One way is easy. suppose $i$ is coprime to $n$ then ...
4
votes
1answer
106 views

no prime numbers in a disc in $\mathbb{C}$ with radius R in $\mathbb{Z}[i]$

Show that there is a disc in $\mathbb{C}$ with radius R , so that no primes of $\mathbb{Z}[i]$ are contained in the disc. I was thinking of taking a disc which which does not touch (0,0), for ...
4
votes
4answers
866 views

Modular Arithmetic question, possibly involving Chinese remainder theorem

'6 professors begin courses of lectures on Monday, Tuesday, Wednesday, Thursday, Friday and Saturday, and announce their intentions of lecturing at intervals of 2,3,4,1,6,5 days respectively. The ...
3
votes
0answers
191 views

Show $\mathrm{E}(\mathbb{Z}_{mn})$ is isomorphic to $\mathrm{E}(\mathbb{Z}_{m})\times\mathrm{E}(\mathbb{Z}_{n})$ if and only if $(m,n)=1$

Define $\mathrm{E}(\mathbb{Z}_{i})$ to be the group of invertible elements of the ring with unity $\mathbb{Z}_{i}$. Show that $\mathrm{E}(\mathbb{Z}_{mn})$ is isomorphic to ...
3
votes
1answer
456 views

Associates in the Ring of Polynomials mod n

Let $a$ and $b$ be elements of the polynomial ring $\mathbb{Z}/n\mathbb{Z}[x]$. If $a$ and $b$ generate the same ideal, must they be associates? If $n$ is prime, then it is easy to see that the answer ...